I wanted to solve a recurrence relation of the form
$$(a+2b+c)f_{n}=bf_{n+1}+bf_{n-1}+cf_{0}$$
I know how to get the solution using the characteristic equation method but I wanted to solve it using Generating functions method. I have used $G(x)=\sum_{n=0}^{\infty}f_{n}{x^n}$ as the generating function. While proceeding for the solution, I reached till the partial fractions for the $G(x)$ and obtained
$$ G(x) = \frac{(\frac{cf_0}{a+c})}{1-x} + \frac{(\frac{bf_0}{a+c})(a+cx)}{bx^2-(a+2b+c)x+b} $$
I know the first term has the power series form of $\sum_{n=0}^{\infty}x^n$ but I cannot figure out how to obtain the power series form for the second term so that I can get the general form of $f_n$.
Also, it is given that $f_m=1$ where $m$ is a positive integer.